Cheryl M. Lam Advisor: Mark A. Cappelli Stanford Plasma Physics Laboratory

1

Dissertation Outline

I. Introduction

II. Background

III. Model Description

IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline

2D SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN ELECTRON TRANSPORT IN A HALL THRUSTER

Stanford UniversityPlasma Physics Lab

AXIAL-AZIMUTHAL HYBRID FLUID-PIC SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN ELECTRON TRANSPORT IN A HALL

THRUSTER

Cheryl M. Lam

Advisor: Mark A. Cappelli

Stanford Plasma Physics Laboratory

Mechanical Engineering Department

Dissertation Proposal Meeting

December 20, 2013

2


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




I. Introduction

II. Hall Thruster Simulations (Background)

III. Model Description: Hybrid Fluid-PIC z-θ Model

IV. Model Sensitivities

V. Simulation Results

VI. Discussion

VII. Conclusions and Future Work

3


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Hall Thruster

Electric space propulsion device Demonstrated high thrust efficiencies

Up to 60% (depending on operating power)

Deployed production technology Design Improvements Better physics understanding

Basic Premise:

Accelerate heavy (positive) ions through electric potential to create thrust E x B azimuthal Hall current

Radial B field (r) Axial E field (z)

Ionization zone (high electron density region) Electrons “trapped” Neutral propellant (e.g., Xe) ionized

via collisions with electrons Plasma

Ions accelerated across imposed axial potential (Ez / Φz) & ejected from thruster

4


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Motivation

Hall thruster anomalous electron transport Super-classical electron mobility observed in experiments1

Theory: Correlated (azimuthal) fluctuations in ne and uez induce super-classical electron transport

2D r-z models use tuned mobility to account for azimuthal effects2,3

3D model is computationally expensive

First fully-resolved 2D z-θ simulations of entire thruster

** Initial development by E. Fernandez

Predict azimuthal (ExB) fluctuations

Quantify impact on electron transport

Channel Diameter = 9 cm

Channel Length = 8 cm

1Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, 026410, 2001. 2Fife, J. M., Ph.D. Dissertation, Massachusetts Inst. of Technology, Cambridge, MA, 1999. 3Fernandez et al, “2D simulations of Hall thrusters,” CTR Annual Research Briefs, Stanford Univ.,1998.

5


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Hall Thruster Simulations

2D radial-axial (r-z) simulations J. M. Fife, 1998 Ph.D. Dissertation – hybrid fluid-PIC E. Fernandez, M. K. Scharfe, 2009 Ph.D. Dissertation – use

experimental/semi-empirical mobility to account for azimuthal effects E. Cha – ongoing – alternate propellants, entropy closure model

2D axial-azimuthal (z-θ) A. K. Knoll, 2010 Ph.D. Dissertation – fluid continuum, predicts high

frequency fluctuations ~1-40 MHz, run length ~10 μs L. Garrigues et al., IEPC 2013, PIC, partial azimuth, grid/timestep

scaling, run length ~40 μs C. M. Lam – ongoing – hybrid fluid-PIC, run length ~200 μs

3D F. Taccogna et al, IEPC 2013, PIC-MCC, geometric scaling and partial

azimuth to reduce computational cost, run length ~5μs K. Matyash, R. Schneider, S. Mazzoufre, Y. Raitses et al, IEPC 2013,

PIC-MCC, partial azimuth, run length ~25 μs

6


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Relevance of Hybrid z-θ Simulations

Thruster geometry

Full-size thruster: Dthruster ≈ 9 cm, Lthruster = 8 cm Axial extent: entire thruster (anode to exit plane) plus near-field plume Resolve full azimuth

No artificial introduction of periodicity No geometric (or grid/timestep) scaling

Time scales of interest Hybrid approach enables longer (~100s μs) simulations Enables study of low- to mid-frequency waves (~10 kHz – 100 MHz)

7


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



2D in z-θ No radial dynamics

E x B + θ

Br: purely radial

(measured from SHT laboratory discharge)

Imposed operating voltage

(based on operating condition)

Geometry

extends 4 cm past channel exit

Channel Diameter = 9 cm

Channel Length = 8 cm

Anode Cathode

G

Anode Exit Plane

GSAMPLE GRID:

z: 40 points, non-uniformθ: 50 points, uniform

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I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Hybrid Fluid-PIC Model

Ions: Particle-In-Cell approach (super-particles) Non-magnetized No particle-particle collisions; Wall collisions modeled in some cases

Neutrals: Particle-In-Cell approach (super-particles) Injected at anode per mass flow rate No particle-particle collisions; Wall collisions modeled in some cases Ionized per local ionization rate

Electron impaction ionization rate based on fits to experimentally-measured collision cross-sections, assuming Maxwellian distribution for electrons

Electrons: Fluid continuum Continuity (species & current) Momentum

Drift-diffusion equation Inertial terms neglected

Energy (1D in z) Convective & diffusive fluxes Joule heating, Ionization losses, Effective wall loss

Quasineutrality:ni = ne

9


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Particle-In-Cell (PIC) Approach Particles: arbitrary positions

Force Particle acceleration

Interpolate: Grid Particle Plasma properties evaluated at grid points

(Coupled to electron fluid solution) Interpolate: Particle Grid

Bilinear Interpolation

Ions subject to electric field:

PIC Ions & Neutrals

rNW

rSE

rNE

rSW

FNW

FSE

FNE

FSW

Interpolation:Particle Grid

Interpolation:Grid Particle

BuqEqamFLorentz

≈ 0

neglect

Discrete particles result in “noisy” plasma properties at grid

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I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Neutral Injection & Particle Collisions

Particle collisions with thruster walls – included for some simulations Neutral particles reflected upon collision with anode or inner/outer

radial channel walls Ions recombine (with donor electron) to form neutral upon collision with

inner/outer radial channel walls Particles still otherwise collisionless, i.e., we do not model particle-

particle collisions

Neutral injection: Injection velocity sampled from half-Maxwellian distribution No wall collisions: mean speed based on centerline (channel radial

midpoint) velocity from r-z simulations With wall collisions: Tanode ~ 1000K

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I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Fluid Equations

Species Continuity

Current Continuity

eeee nunt

n

)(

0

Jt

0

ni = ne

12


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




Momentum: Drift-Diffusion Neglect inertial terms

ue E Dner

ne

1

1 en

ce

2

Ez

Br 1

1 en

ce

2kTe

eneBr

ne

z 1

1 en

ce

2k

eBr

Te

z

)1( 2

2

en

ceenm

e

Classical Mobility

e

kTD e

uez Ez Dne

ne

z D

Te

Te

z 1

1 en

ce

2

EBr 1

1 en

ce

2

kTe

eneBrrne

Classical Diffusion

13


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline





ue E Dner

ne

1

1 en

ce

2

Ez

Br 1

1 en

ce

2kTe

eneBr

ne

z 1

1 en

ce

2k

eBr

Te

z

)1( 2

2

en

ceenm

e

Classical Mobility

e

kTD e

uez Ez Dne

ne

z D

Te

Te

z 1

1 en

ce

2

EBr 1

1 en

ce

2

kTe

eneBrrne

θ fluctuations/dynamics

classical E x B diamagnetic

Classical Diffusion


14


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




Combine current continuity and electron momentum to get convection-diffusion equation for Φ:

A1

2 2 A2

A3

2z2 A4

z A5 0, where

E

where (φ is electric potential)

A1 ne

r2, A3 ne ,

A2 1r

( ne

r

rne

1

1 en

ce

2

z

ne

Br

ne

Br

z

1

1 en

ce

2 )

A4 1

1 en

ce

2

1rBr

ne

ne

z

ne

z

ne

rBr

1

1 en

ce

2

A5 f (ne ,Te ,, en ,ce ) ne

rui

ui

rne

ne

uiz

z uiz

ne

z

15


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline




Energy (Temperature) Equation 1D in z (average over θ, then time advance 1D equation in z)

wallionizjouleeeeeeee

e SSSqukTnTut

Tkn

)(23

eeneejoule unmS

eeieiioniz kTnEnS2

3

23

1

2eewall

wwallwall Tne

kTnS

where

with ionization cost factor αi = 1 (simplest model)

neglect shape factor variation in z, other simplifying assumptions

walln

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I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Ionization Loss

Simplest model: Use constant ionization cost factor

Dugan model: temperature-dependent ionization cost factor

eeieiioniz kTnEnS2

3

2677.0

exp254.0

e

ii kT

E

αi = 1 – 2.5 (up to ~5)

17


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Solution Algorithm

Iterative Solve Φ

Time Advance Particle Positions & VelocitiesNeutrals & Ions (subject to F=qE)

Ionize Neutrals

Inject Neutrals

Calculate Plasma Propertiesni-PART, vi-PART, nn-PART, vn-PART ni-GRID, vi-GRID, nn-GRID, vn-GRID

QUASINEUTRALITY: ne = ni = nplamsa

Time Advance Te=Te(ne, ve)

Calculate Φ=Φ(ne, vi-GRID) ↔ EGRID

Calculate ve=ve(Φ, ne, Te)

r = Φ – Φlast-iterationr < ε0

CONVERGED

Calculate vi-GRID-TEST= vi-GRID(EGRID)

EGRID EPART

LEAPFROG

RK4

DIRECT SOLVE 2nd-order F-D

Spline

Boundary Conditions:

• Dirichlet in z (Φ,Te)

• Periodic in θ

18


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Numerical Solution

Single numerical grid used for PIC and fluid solution

Cubic spline applied to PIC-derived grid properties (prior to use in fluid equations)

Electron energy (Te) equation Central difference scheme for spatial derivatives Calculate RHS for 2D grid, then average over θ to obtain 1D Te(z) Time advance via 4th-order Runge-Kutta

Electric potential (Φ) equation 2nd-order finite difference w/ upwind Direct solve: block tridiagonal solver

Single timestep used for PIC and fluid (typically dt = 1 ns)

19


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Sensitivities

Grid spacing Current non-conservation Predicted waves (azimuthal modes) Effect of spline / PIC “noise” – required number of particles?

Initial Conditions

Boundary Conditions

Numerical stability/sensitivity of energy (Te) equation

Ionization cost factor Constant factor Dugan model

Energy loss to wall

20


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Sample Numerical Grid

40 points non-uniform in z50 points uniform in θ

61 points uniform in z25 points uniform in θ

~400,000 (initial) particles per speciesdt = 1ns

6 days to run ~200 μs (single 64-bit Xeon x5355 2.66GHz processor core)

21


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Sample ResultsIEPC 2009

Simulation Details• No wall collisions modeled• Ionization cost factor = 1

Initial Conditions• Uniform # particles per cell• Inverted Maxwellian velocity distribution (particles)

Operating Voltage

Predicted Current

100V

1-2 A

Neutral Injection 2 mg/s

Grid40 points (non-uniform) in z

50 pints (uniform) in θ

Timestep

Run Length

dt = 1 ns

200 μs

Computational Performance

6 days on Intel Xeon 5355 2.66 GHz (64-bit single core)

22


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Plasma Density

Time-Averaged Plasma Properties100 V Simulation – IEPC 2009

Electron Temperature

Axial Ion Velocity Potential

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I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



E x B

Axial Electron Velocity

Distinct wave behavior observed:

Throughout channel (upstream) Tilted: - z, + E x B Lower frequency, slow moving,

longer wavelength

Near exit plane

Peak Br, High shear (∂ueθ/∂z) Tilted: + z, - ExB Higher frequency, faster moving,

shorter wavelength

Outside exit plane (downstream) Purely axial: + z Same structure (in θ) as exit

plane waves

24


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Fluctuations in θ

Anode Cathode

E x B

E x B

E x B

f = 40 KHzλθ = 5 cmvph = 4000 m/s

f = 700 KHz

λθ = 4 cmvph = 40,000 m/s

25


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Electron Transport

Simulation predicts super-classical electron mobility

Axial Electron Mobility:ze

ez

Eqn

J

26


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Stability Challenges

100V simulation (2013) Ionization cost factor = 2.1 Wall collisions modeled ICs: smooth neutral and ion density profiles, experimental Te(z) Strong instability develops after ~100 μs (dt = 1 ns)

27


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Stability Challenges

100V simulation (2013) Ionization cost factor = 2.1 Wall collisions modeled ICs: smooth neutral and ion density profiles, experimental Te(z) Strong instability develops after ~100 μs (dt = 1 ns)

28


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Runaway Ionization

100 V (2013)

29


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Current Non-Conservation100V simulation (2013)

30


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Theory:Azimuthal Fluctuations induce Axial Transport

Consider

Induced Current

r

ce

en

ez B

Eu

xBE

2

1

1

xBExBE ezeez uqnJ

cos2

1

)cos(

1

1)cos(

200

0

020

T

v

En

B

qJ

dttB

EtnqJ

ce

enr

eez

T

tr

ce

en

eez

xBE

xBE

Induced current depends on phase shift ξ

t

ξ

Eθ = E0cos(ωt)

ne = n0cos(ωt + ξ)

31


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline





Correlated azimuthal fluctuations induce axial transport:

ue E Dner

ne

1

1 en

ce

2

Ez

Br 1

1 en

ce

2kTe

eneBr

ne

z 1

1 en

ce

2k

eBr

Te

z

)1( 2

2

en

ceenm

e

Classical Mobility

e

kTD e

uez Ez Dne

ne

z D

Te

Te

z 1

1 en

ce

2EBr 1

1 en

ce

2kTe

eneBrrne

Previous modelsunder-predict

Jez=qneuezθ fluctuations/dynamics

eeinducede unJ~~

,


Classical Diffusion


32


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Correlated ne and uez fluctuations generate axial electron current

Correlated fluctuations generate axial current

Uncorrelated

100 V (2013)

33


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Anomalous Electron Transport Characterize fluctuations

Compare to experimental data Consider including dispersion

analysis/maps Compare to theory or linearized

dispersion relations?

Role of fluctuations in enhanced (anomalous) electron transport

Effect of shear, gradients, etc. on anomalous transport

Effect of operating conditions

100 V (2009)

34


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Sample Experiments

35


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Take-Aways

Simulations predict fluctuations Complementary to other simulation efforts Similar to those observed in experiment? Consistent with theory?

Anomalous electron transport Role of fluctuations Effect of Hall thruster geometry, operating conditions, etc.

Suggestions for future work Finite volume (in process) Fully kinetic simulations

36


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Recent Progress & Challenges

Addition of particle collisions with thruster walls

Finer axial (z) grid resolution near anode

Stability challenges Sensitivity to Initial Conditions and Boundary Conditions Strong fluctuation in Te and Φ Current conservation

Finite Difference – present model Finite Volume – parallel effort (E. Fernandez)

37


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Additional Simulations – 100V

Establish stable long-running simulation (~600 μs – 1 ms) for low voltage (100V) case Start (continue) from IEPC 2009 simulation (run length = 200 μs)

Ionization cost factor = 1 No wall collisions; Slow neutral injection velocity Zero-slope BC for Te

Increase number of particles (ionizspc) to enable longer simulation

Grid refinement study Finer grid in z: current non-conservation, wave structure near anode Finer, varied grid in θ: impact periodicity, azimuthal wavelength/modes

Initial Conditions Increased neutral density resulting spoke at anode? Shape of neutral density profile (flat vs. sloped, magnitude/gradient) More realistic (experiment-like) plasma density profile and/or

magnitude

38


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Additional Simulations

Higher voltage Incrementally increase operating voltage Look for trends (in frequency/wavelength/direction of fluctuations,

electron transport, anomalous contribution to transport)

Initial Conditions Waves Smooth initial profiles (based on prescribed profile or experiment)

allow fluctuations to evolve Consider “seeding” simulation with particular waves (spatial modes) to

study wave growth/dissipation and energy coupling (e.g., into other modes)

39


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Improvements

Te stability and BC/IC impacts Stability and sensitivity analysis – contribution of source/sink terms,

esp. wall loss and ionization cost Enforcement of experimental profile (as IC and/or prescribed profile at

regular time interval) and/or experimental-based limits Prescribed (fixed value) vs. zero-slope condition at axial domain

boundaries Ionization cost factor

Dugan model – exponential vs. algebraic form Tuned constant factor?

Improved/tunable wall loss model Introduction of diffusive damping term? Effect of spline smoothing Implicit solve 2D Te equation

Improve stability – consider more global changes to model “External” power supply circuit model (potential BC) Hyperviscous damping (for potential equation)

40


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Model Improvements

Incremental changes to PIC model – additional physics Introduction of wall collisions (w/ higher neutral injection velocity) Revisit ionization rate implementation

Enhance electron transport via prescribed electron mobility – sustain/generate waves (may also improve stability)

Additive “baseline” μ┴ or νen

Experimental mobility μexp(z) (in lieu of or in addition to μ┴) Experimental or additional (or Bohm-like) mobility for electron fluid

equations only

41


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Publications

Conference Papers C. M. Lam, A. K. Knoll, E. Fernandez, and M. A. Cappelli, “Two-

Dimensional (z-θ) Simulations of Hall Thruster Anomalous Transport,” International Electric Propulsion Conference, 2009.

C. M. Lam, E. Fernandez, and M. A. Cappelli, “Two-Dimensional (z-θ) Hybrid Fluid-PIC Simulation of Enhanced Cross-field Electron Transport in an Annular E x B Discharge,” Gaseous Electronics Conference, 2012.

C. M. Lam, E. Fernandez, and M. A. Cappelli, “Two-Dimensional Simulations of Coherent Fluctuation-Driven Transport in a Hall Thruster,” International Electric Propulsion Conference, 2013.

Journal Papers C. M. Lam, E. Fernadez, and M. A. Cappelli, “A Two-Dimensional

Hybrid Hall Thruster Simulation that resolves the E × B Electron Drift Direction,” IEEE Transactions on Plasma Science, Special Edition – submitted for review, expect publication Dec 2014

Planned additional publications Journal paper: waves and transport

42


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Research Progress Timeline

MILESTONE TARGET

Additional simulations – 100 V

early Jan

Stability improvements

Sensitivity studies

Consider additional physics, as warranted

end Jan

(2-3 wk delay)

Higher voltage simulations

mid Feb

Simulation results analysis

end Feb

DISSERTATION TARGET

I. Introduction mid Jan

II. Background mid Jan

III. Model Description: Hybrid Fluid-PIC z-θ Model

end Jan

IV. Model Sensitivities end Feb

V. Simulation Results mid Mar

VI. Discussion end Mar

VII. Conclusions end Mar

TARGET – Final Draft: mid April 2014

43


I. Introduction

II. Background


IV. Sensitivities

V. Results

VI. Discussion

VII. Conclusions

Research Plan

Timeline



Questions?

Documents

Cheryl M. Lam Advisor: Mark A. Cappelli Stanford Plasma Physics Laboratory